aa r X i v : . [ m a t h . N T ] D ec REFINED CLASS NUMBER FORMULAS FOR G m BARRY MAZUR AND KARL RUBIN
Abstract.
We formulate a generalization of a “refined class number formula”of Darmon. Our conjecture deals with Stickelberger-type elements formed fromgeneralized Stark units, and has two parts: the “order of vanishing” and the“leading term”. Using the theory of Kolyvagin systems we prove a large partof this conjecture when the order of vanishing of the corresponding complex L -function is 1. Contents
1. Introduction 12. Unit groups 23. A Stark conjecture over Z
34. The Artin regulator 55. The conjecture 76. Order of vanishing 97. The case K = k r = 1 1611. Evidence in the case of general r Introduction
In [D], Darmon conjectured a “refined class number formula” for real quadraticfields, inspired by work of Gross [G1], of the first author and Tate [MT], andof Hayes [H]. The common setting for these conjectures included a finite abelianextension
L/K and a Stickelberger-type element θ ∈ Z [Gal( L/K )]. In analogy withthe Birch and Swinnerton-Dyer conjecture, these conjectures predicted the “orderof vanishing” (a nonnegative integer r such that θ lies in the r -th power of theaugmentation ideal A of Z [Gal( L/K )]) and the “leading term” (the image of θ in A r / A r +1 ) of θ .In [MR3], we proved most (the “non-2-part”) of Darmon’s conjecture, usingthe theory of Kolyvagin systems [MR1]. The key idea is that in nice situations, Date : December 17, 2013.2010
Mathematics Subject Classification.
Primary 11R42, 11R27; Secondary 11R23, 11R29.This material is based upon work supported by the National Science Foundation under grantsDMS-1302409 and DMS-1065904. the space of Kolyvagin systems is a free Z p -module of rank one, and hence twoKolyvagin systems that agree “at n = 1” must be equal. Darmon’s conjecture for n = 1 follows from the classical evaluation of L ′ (0 , χ ) for a real quadratic Dirichletcharacter χ .In this paper we attempt to generalize both the statement and proof of Darmon’sconjecture. To generalize the statement we rely on a suitable version of Stark’s con-jectures. Namely, given a finite abelian tower of number fields L/K/k , our proposedConjecture 5.2 relates the so-called “Rubin-Stark” elements ǫ L,S L attached to L/k (see §
3) with an “algebraic regulator” (see Definition 4.5) constructed from Rubin-Stark elements ǫ K,S L attached to K/k and L . Similar generalizations of Darmon’sconjecture have recently been proposed independently by Sano [S1, Conjecture 4]and Popescu [P2].Our conjecture has two parts, the “order of vanishing” and the “leading term”.We prove a large portion of the order of vanishing part of the conjecture in Theorem6.3. We prove a large part of the leading term statement in Theorem 10.7 followingthe method of [MR3], but only under the rather strong assumption that the orderof vanishing (the “core rank”, in the language of [MR4]) is one. As L varies, theelements ǫ L,S L form an Euler system, and the elements ǫ K,S L form what we call aStark system. When the order of vanishing is one we can relate these systems andprove the leading term formula. In the final section we prove a weakened versionof the leading term statement for general r , under some additional hypotheses. Notation.
Suppose throughout this paper that O is an integral domain with fieldof fractions F , and let R = O [Γ] with a finite abelian group Γ. We are mainlyinterested in the case where O = Z or Z p for some prime p .If M is an R -module, we let M ∗ := Hom R ( M, R ). If ρ ∈ R , then M [ ρ ] willdenote the kernel of multiplication by ρ in M .If r ≥
0, then ∧ r M (or ∧ rR M , if we need to emphasize the ring R ) will denotethe r -th exterior power of M in the category of R -modules, with the conventionthat ∧ M = R . See Appendix A for more on the exterior algebra that we use. Inparticular, in Definition A.3 we define an R -lattice ∧ r, M ⊂ ∧ r M ⊗ F , containingthe image of ∧ r M , that will play an important role.2. Unit groups
Suppose
K/k is a finite abelian extension of number fields. Let Γ = Gal(
K/k )and R = Z [Γ]. Fix a finite set S of places of k containing all infinite places and allplaces ramified in K/k , and a second finite set T of places of k , disjoint from S .Define: S K = { places of K lying above places in S } ,T K = { places of K lying above places in T } ,U K,S,T = { x ∈ K × : | x | w = 1 for all w / ∈ S K , x ≡ w ) for all w ∈ T K } . We assume further that K has no roots of unity congruent to 1 modulo all placesin T K , so that U K,S,T is a free Z -module. When there is no fear of confusion, wewill suppress the S and T and write U K := U K,S,T
Suppose now that L is a finite abelian extension of k containing K . Let G :=Gal( L/k ) and H := Gal( L/K ), so
G/H = Γ. Let A H ⊂ Z [ H ] be the augmentationideal, the ideal generated by { h − h ∈ H } . EFINED CLASS NUMBER FORMULAS FOR G m Corollary 2.1.
For every s ≤ r and every ρ ∈ Q [Γ] , Proposition A.6 gives acanonical pairing ( ∧ r, U K )[ ρ ] × ∧ r − s Hom Γ ( U K , Z [Γ] ⊗ Z A H / A H ) −→ ( ∧ s, U K )[ ρ ] ⊗ Z A r − sH / A r − s +1 H . Proof.
Apply Proposition A.6 with B := ⊕ i ≥ A iH / A i +1 H and n = 1. (cid:3) A Stark conjecture over Z In this section we recall the so-called Rubin-Stark conjecture over Z for arbitraryorder of vanishing from [R1]. When the order of vanishing (the integer r below) isone, this is essentially the “classical” Stark conjecture over Z (see for example [T, § IV.2] and [R1, Proposition 2.5]).Keep the finite abelian extension
K/k of number fields from §
2, with Γ =Gal(
K/k ), and the sets
S, T of places of K . We define the Stickelberger functionattached to K/k (and S and T ) to be the meromorphic C [Γ]-valued function θ K/k ( s ) = θ K/k,S,T ( s ) = Y p / ∈ S (1 − Fr − p N p − s ) − Y p ∈ T (1 − Fr − p N p − s )where Fr p ∈ Γ is the Frobenius of the (unramified) prime p . If χ ∈ ˆΓ := Hom(Γ , C × ),then applying χ to the Stickelberger function yields the (modified at S and T ) Artin L -function χ ( θ K/k ( s )) = L S,T ( K/k ; ¯ χ, s ) . Definition 3.1. If w is a place of K we write K w for the completion of K at w and | | w : K w → R + ∪ { } for the absolute value normalized so that | x | w = ± x (the usual absolute value) if K w = R ,x ¯ x if K w = C , N w − ord w ( x ) if K w is nonarchimedeanwhere N w is the cardinality of the residue field of the finite place w . Definition 3.2.
Suppose now that S ′ ⊂ S is a subset such that every v ∈ S ′ splits completely in K/k . Let r = | S ′ | ≥
0. Let S ′ K denote the set of primes of K above S ′ , and let W K,S ′ denote the free abelian group on S ′ K , so W K,S ′ is a free Z [Γ]-module of rank r .Define a Z [Γ]-homomorphism η log K : U K → W K,S ′ ⊗ R by η log K ( u ) = X w ∈ S ′ K w ⊗ log | u | w . If L is an abelian extension of K with Galois group H := Gal( L/K ), and A H ⊂ Z [ H ] is the augmentation ideal, let [ · , L w /K w ] : K × w → H denote the local Artinsymbol (this is independent of the choice of place of L above w ) and define a Z [Γ]-homomorphism η Art
L/K : U K → W K,S ′ ⊗ Z A H / A H by η Art
L/K ( u ) := X w ∈ S ′ K w ⊗ ([ u, L w /K w ] − . Definition 3.3.
Let R ∞ = R ∞ K,S,T,S ′ : ∧ r U K ⊗ Γ ∧ r W ∗ K,S ′ −→ R [Γ]be the classical regulator map induced by η log K : ∧ r U K → ∧ r W K,S ′ ⊗ R and thenatural isomorphism ∧ r W K,S ′ ⊗ ∧ r W ∗ K,S ′ → Z [Γ]. BARRY MAZUR AND KARL RUBIN
Concretely, the map R ∞ is given as follows. If w ∈ S ′ K , let w ∗ ∈ W ∗ K,S ′ be themap w ∗ (cid:18) X z ∈ S ′ K a z z (cid:19) := X γ ∈ Γ a γw γ. If v , . . . , v r is an ordering of the places in S ′ , and for each i we choose a place w i of K above v i , then w ∧ · · · ∧ w r is a Z [Γ]-basis of ∧ r W K,S ′ , and w ∗ ∧ · · · ∧ w ∗ r isthe dual basis of ∧ r W ∗ K,S ′ . Then R ∞ (( u ∧ · · · ∧ u r ) ⊗ ( w ∗ ∧ · · · ∧ w ∗ r )) = det (cid:0)X γ ∈ Γ log | u γi | w j γ − (cid:1) . Definition 3.4.
Write for the trivial character of Γ. For every χ ∈ ˆΓ there is anidempotent e χ = | Γ | − X γ ∈ Γ χ ( γ ) γ − ∈ C [Γ] , and we define a nonnegative integer r ( χ ) = r ( χ, S ) by(3.5) r ( χ ) = ord s =0 L S ( s, ¯ χ ) = dim C e χ C U K = ( |{ v ∈ S : χ (Γ v ) = 1 }| if χ = | S | − χ = where Γ v is the decomposition group of v in Γ (see for example [T, PropositionI.3.4]). If r ≥ S contains r places that split completely in K/k , and | S | ≥ r + 1, then r ( χ ) ≥ r for every χ ∈ ˆΓ, and we let ρ K,r := X χ ∈ ˆΓ ,r ( χ ) = r e χ ∈ Q [Γ] . The following is the “Stark conjecture over Z ” that we will use. Conjecture
St(
K/k, S, T, S ′ ) (= Conjecture B ′ of [R1]) . Suppose that: (i) S is a finite set of places of k containing all archimedean places and allplaces ramifying in K/k , (ii) T is a finite set of places of K , disjoint from S , such that U K,S,T containsno roots of unity, (iii) S ′ ( S contains only places that split completely in K .Let r = | S ′ | . Then there is a unique element ǫ K = ǫ K,S,T,S ′ ∈ ( ∧ r, U K,S,T )[ ρ K,r ] ⊗ Γ ∧ r W ∗ K,S ′ such that R ∞ ( ǫ K ) = lim s → s − r θ K/k ( s ) . By Conjecture St(
K/k ) we will mean the conjecture that St(
K/k, S, T, S ′ ) holdsfor all choices of S , T , and S ′ satisfying the hypotheses above.Recall that ∧ r W ∗ K,S ′ is free of rank one over Z [Γ]. The uniqueness of ǫ K,S,T,S ′ is automatic because R ∞ is injective on ( ∧ r, U K,S,T )[ ρ K,r ] ⊗ Γ ∧ r W ∗ K,S ′ (see forexample [R1, Lemma 2.7]).Conjecture St( K/k, S, T, S ′ ) is known to be true in the following cases: • r = 0 (in which case ǫ K := θ K/k (0) ∈ Z [Γ], which was proved indepen-dently by Deligne and Ribet, Cassou-Nogu`es, and Barsky), • K/k is quadratic ([R1, Theorem 3.5]),
EFINED CLASS NUMBER FORMULAS FOR G m • k = Q (proved by Burns in [Bur, Theorem A]), • S − S ′ contains a prime that splits completely in K/k ([R1, Proposition3.1]).
Lemma 3.6.
Suppose that S − S ′ contains a place that splits completely in K/k ,and | S − S ′ | ≥ . Then ǫ K = 0 satisfies Conjecture St(
K/k, S, T, S ′ ) .Proof. In this case r ( χ, S ) > r = | S ′ | for every χ ∈ ˆΓ, so lim s → s − r θ K/k ( s ) = 0and ρ K,r = 1 in Definition 3.4 . The lemma follows. (cid:3) The Artin regulator
Fix a finite abelian extension
L/k of number fields, and an intermediate field K , k ⊂ K ⊂ L . Let G := Gal( K/k ), H := Gal( K/F ) and Γ := Gal(
F/k ) =
G/H . LK H ✓✓✓✓✓✓✓✓ k Γ ⑥⑥⑥⑥⑥ G Fix a finite set S of places of k containing all archimedean places and all primesramifying in L/k . Fix a second finite set of primes T of k , disjoint from S , suchthat U L = U L,S,T contains no roots of unity.Suppose that we have a filtration S ′ ⊂ S ′′ ( S , where every v ∈ S ′′ splitscompletely in K/k , and every v ∈ S ′ splits completely in L/k . Let r = | S ′ | and s = | S ′′ | − | S ′ | .For the rest of this section, we keep S, S ′ , S ′′ and T fixed, and we suppress themfrom the notation when possible.For every subset Σ ⊂ S ′′ , let W K, Σ denote the free abelian group on the set ofprimes of K above Σ, and similarly with L in place of K . Then W K, Σ is a free Z [Γ]-module of rank | Σ | , we have(4.1) W K,S ′′ = W K,S ′ ⊕ W K,S ′′ − S ′ , ∧ r + s W K,S ′′ = ∧ r W K,S ′ ⊗ Γ ∧ s W K,S ′′ − S ′ , and the natural map S L → S K , that takes a place of L to its restriction to K ,induces an isomorphism of free modules(4.2) W L,S ′ ⊗ G Z [Γ] ∼ −−→ W K,S ′ . Let η Art
L/K ∈ Hom Γ ( U K , W K,S ′′ − S ′ ⊗ Z A H / A H ) be the map of Definition 3.2, withthe augmentation ideal A H as in §
2. Composition with η Art
L/K gives a Z [Γ]-homo-morphism(4.3) W ∗ K,S ′′ − S ′ −→ Hom Γ ( U K , Z [Γ] ⊗ Z A H / A H ) . Corollary 2.1 gives a canonical pairing( ∧ r + s, U K ) × ∧ s Hom Γ ( U K , Z [Γ] ⊗ Z A H / A H ) −→ ( ∧ r, U K ) ⊗ Z A sH / A s +1 H , and using (4.3) we can pull this back to a pairing(4.4) ( ∧ r + s, U K ) × ∧ s W ∗ K,S ′′ − S ′ −→ ( ∧ r, U K ) ⊗ Z A sH / A s +1 H . BARRY MAZUR AND KARL RUBIN
Definition 4.5.
Tensoring both sides of (4.4) with ∧ r W ∗ K,S ′ and using (4.1), wedefine an algebraic regulator map R Art
L/K = R Art
L/K,S,S ′ ,S ′′ R Art
L/K : ( ∧ r + s, U K ) ⊗ Γ ∧ r + s W ∗ K,S ′′ −→ ( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ ⊗ Z A sH / A s +1 H . Definition 4.6.
Let ι L/K : Z [Γ] ֒ → Z [ G ] denote the natural Z [ G ]-module homo-morphism that sends γ ∈ Γ to P g ∈ γ g , viewing γ as an H -coset. Then ι L/K is nota ring homomorphism, but rather(4.7) ι L/K ( α ) ι L/K ( β ) = [ L : K ] ι L/K ( αβ ) . Note that ι L/K is a Z [ G ]-module isomorphism Z [Γ] ∼ −→ Z [ G ] H .As in §
2, let U ∗ K := Hom Γ ( U K , Z [Γ]) , U ∗ L := Hom G ( U L , Z [ G ]) . If ϕ ∈ U ∗ L , then ϕ ( U K ) ⊂ Z [ G ] H , and we define ϕ K = ι − L/K ◦ ϕ | U K ∈ U ∗ K .Let j L/K : U K ֒ → U L denote the natural inclusion, and ∧ s j L/K : ∧ s U K → ∧ s U L the induced map (if s = 0, we let ∧ j L/K = ι L/K : Z [Γ] → Z [ G ]). Lemma 4.8. ∧ r j L/K ( ∧ r, U K ) ⊂ [ L : K ] max { ,r − } ∧ r, U L .Proof. If r = 0 there is nothing to prove, so assume r ≥
1. Suppose ϕ , . . . , ϕ r ∈ U ∗ L .Let ϕ = ϕ ∧ · · · ∧ ϕ r ∈ ∧ r U ∗ L , and ϕ K = ϕ K ∧ · · · ∧ ϕ Kr ∈ ∧ r U ∗ K . Using (4.7)and the evaluation (A.2) of ϕ and ϕ K as determinants, we have a commutativediagram ∧ r, U L (cid:31) (cid:127) / / ∧ r U L ⊗ Q ϕ / / Q [ G ] ∧ r, U K (cid:31) (cid:127) / / ∧ r U K ⊗ Q ϕ K / / ∧ r j L/K O O Q [Γ] [ L : K ] r − ι L/K O O By definition ϕ K ( ∧ r, U K ) ⊂ Z [Γ], so ϕ ( ∧ r j L/K ( ∧ r, U K )) ⊂ [ L : K ] r − Z [ G ]. Sincethese ϕ generate ∧ r U ∗ L , this proves the lemma. (cid:3) Lemma 4.9.
The map [ L : K ] − max { ,r − } ∧ r j L/K : ∧ r, U K → ∧ r, U L and theinverse of the isomorphism (4.2) induce a map j L/K : ( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ −→ ( ∧ r, U L ) ⊗ G ∧ r W ∗ L,S ′ . Proof.
Using (4.2) for the second equality, we have( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ = ( ∧ r, U K ) ⊗ G ∧ r W ∗ K,S ′ = ( ∧ r, U K ) ⊗ G ( ∧ r W ∗ L,S ′ ⊗ G Z [Γ])= (( ∧ r, U K ) ⊗ G Z [Γ]) ⊗ G ∧ r W ∗ L,S ′ = ( ∧ r, U K ) ⊗ G ∧ r W ∗ L,S ′ . Now the lemma follows from Lemma 4.8. (cid:3)
EFINED CLASS NUMBER FORMULAS FOR G m The conjecture
Let
L/K/k , G , H , Γ, S , T , S ′ , S ′′ be as in §
4. The hypotheses of ConjecturesSt(
L/k, S, T, S ′ ) and St( K/k, S, T, S ′′ ) are all satisfied, and if those conjectures bothhold they provide us with elements ǫ L := ǫ L,S,T,S ′ ∈ ( ∧ r, U L )[ ρ L,r ] ⊗ G ∧ r W ∗ L,S ′ ⊂ ∧ r U L ⊗ G ∧ r W ∗ L,S ′ ⊗ Z Q ,ǫ K := ǫ K,S,T,S ′′ ∈ ( ∧ r + s, U K )[ ρ K,r + s ] ⊗ Γ ∧ r + s W ∗ K,S ′′ ⊂ ∧ r + s U K ⊗ Γ ∧ r + s W ∗ K,S ′′ ⊗ Z Q . Definition 5.1. If M is a Z [ H ]-module, define the twisted trace Tw L/K : M −→ M ⊗ Z Z [ H ]by Tw L/K ( m ) := P h ∈ H m h ⊗ h − ∈ M ⊗ Z Z [ H ].We will think of Tw L/K ( ǫ L ) as a generalized Stickelberger element. The followingconjecture is inspired by conjectures in [MT, G1, G2, D]. Conjecture 5.2.
With ( L/K/k, S, T, S ′ , S ′′ ) as in §
4, suppose that Conjectures
St(
L/k, S, T, S ′ ) and St(
K/k, S, T, S ′′ ) both hold. (i) “Order of vanishing”: Tw L/K ( ǫ L ) ∈ ( ∧ r, U L ) ⊗ G ∧ r W ∗ L,S ′ ⊗ Z A sH . (ii) “Leading term”: with the maps j L/K of Lemma 4.9 and R Art
L/K of Definition4.5, we have Tw L/K ( ǫ L ) ≡ ( j L/K ⊗ R Art
L/K ( ǫ K )) in ( ∧ r, U L ) ⊗ G ∧ r W ∗ L,S ′ ⊗ Z A sH / A s +1 H . Remark 5.3.
Suppose that k = Q , K is a real quadratic field, and L = K ( µ n ) + (the real subfield of the extension of K generated by the n -th roots of unity) with n prime to the conductor of K/ Q . Let S ′ := {∞} and S ′′ := {∞} ∪ { ℓ : ℓ | n } (so r = 1). In this case St( L/k, S, T, S ′ ) and St( K/k, S, T, S ′′ ) are known to hold, andConjecture 5.2 is essentially the same as Darmon’s conjecture in [D, § §
10 for more about the case r = 1. Proposition 5.4. If r = 0 then Conjecture 5.2 is equivalent to the conjecture ofGross in [G1, Conjecture 7.6] and [G2] (see Conjecture ˜ A Z ( L/K/k, S, T, s ) of [P1] ). Before proving Proposition 5.4, we have the following two lemmas. Let J H := Z [ G ] A H be the kernel of the natural projection Z [ G ] ։ Z [Γ]. Lemma 5.5.
There are natural isomorphisms (i) H ∼ −→ A H / A H , given by h ( h − , (ii) Z [Γ] ⊗ Z A rH / A r +1 H ∼ −→ J rH /J r +1 H for every r ≥ , given by γ ⊗ α α ¯ γ ,where ¯ γ is any lift of γ to Z [ G ] .Proof. This is a standard exercise. (cid:3)
Lemma 5.6.
Define ψ : Z [ G ] → Z [ G ] ⊗ Z Z [ H ] by ψ ( ρ ) = P h ∈ H hρ ⊗ h − . Then: (i) ψ is an injective Z [ G ] -module homomorphism (with G acting on the lefton Z [ G ] ⊗ Z Z [ H ] ), BARRY MAZUR AND KARL RUBIN (ii) ψ ( hρ ) = ψ ( ρ ) h for every h ∈ H , (iii) ψ ( J tH ) ⊂ Z [ G ] ⊗ Z A tH for every t ≥ , (iv) for every t ≥ there is a commutative diagram J tH /J t +1 H (cid:23) w ψ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Z [Γ] ⊗ Z A tH / A t +1 H (cid:31) (cid:127) ι L/K ⊗ / / ∼ = O O Z [ G ] ⊗ Z A tH / A t +1 H where the vertical map is the isomorphism of Lemma 5.5(ii).Proof. The first two assertions are clear, and (iii) follows from (ii).To check the commutativity of the diagram in (iv), take γ ∈ Γ and α ∈ A tH .Using (ii), the image of γ ⊗ α in Z [ G ] ⊗ Z A tH / A t +1 H by the upper path is ψ ( α ¯ γ ) = X h ∈ H hα ¯ γ ⊗ h − = X h ∈ H ¯ γh ⊗ αh − , where ¯ γ is any lift of γ to G . The image of γ ⊗ α by the lower path is P h ∈ H ¯ γh ⊗ α .Since α ( h − − ∈ A t +1 H for every h , these are equal in Z [ G ] ⊗ Z A tH / A t +1 H . Thisshows that the diagram in (iv) commutes, and the injectivity of the map inducedby ψ now follows from the injectivity of ι L/K . (cid:3) Proof of Proposition 5.4.
Let ψ be as in Lemma 5.6. If r = 0, then ǫ L = θ L/k (0), soTw
L/k ( ǫ L ) = ψ ( θ L/k (0)). Thus by Lemma 5.6(iii,iv), Conjecture 5.2 is equivalentin this case to the assertions(i) θ L/k (0) ∈ J sH ,(ii) θ L/k (0) ≡ R
Art
L/K ( ǫ K ) (mod J s +1 H ),where we view R Art
L/K ( ǫ K ) ∈ J s +1 H via the isomorphism of Lemma 5.5(ii). This isthe usual statement of Gross’ conjecture [P1, Conjecture ˜ A Z ( L/K/k, S, T, s )]. (cid:3)
Proposition 5.7. If s = 0 (i.e., if S ′′ = S ′ ), then Conjecture 5.2 is true.Proof. Conjecture 5.2(i) is vacuous when r = 0, since by definitionTw L/K ( ǫ L ) ∈ ( ∧ r, U L ) ⊗ Γ ∧ r W ∗ S ′ ⊗ Z Z [ H ] = ( ∧ r, U L ) ⊗ Γ ∧ r W ∗ S ′ ⊗ Z A H . Let N H = P h ∈ H h . In ( ∧ r, U L ) ⊗ Γ ∧ r W ∗ S ′ ⊗ Z [ H ] / A H we have(5.8) Tw L/K ( ǫ L ) = X h ∈ H ǫ hL ⊗ h ≡ X h ∈ H ǫ hL ⊗ N H ǫ L ) ⊗ . If r = 0, then since the image of the Stickelberger element θ L/k (0) under therestriction map Z [ G ] ։ Z [Γ] is θ K/k (0), we have N H ǫ L = N H θ L/k (0) = ι L/K θ K/k (0) = j L/K ( ǫ K ) . By (5.8), this proves Conjecture 5.2(ii) when r = 0.Suppose now that r >
0. Fix generators w L = w ∧· · ·∧ w r and w ∗ L = w ∗ ∧· · ·∧ w ∗ r of ∧ r W L,S ′ and ∧ r W ∗ L,S ′ , respectively. Let w K and w ∗ K be the correspondinggenerators of ∧ r W K,S ′ and ∧ r W ∗ K,S ′ obtained by restricting the w i to K (note thatsince s = 0, we have S ′ = S ′′ ). Choose u L ∈ ∧ r, U L and u K ∈ ∧ r, U K such EFINED CLASS NUMBER FORMULAS FOR G m that ǫ L = u L ⊗ w ∗ L and ǫ K = u K ⊗ w ∗ K . Then [R1, Proposition 6.1] shows that( N H ) r u L = ( ∧ r j )( u K ), and so we also have(5.9) N H ǫ L = [ L : K ] − r ( N H ) r ǫ L = ([ L : K ] − r ( N H ) r u L ) ⊗ w ∗ L = ([ L : K ] − r ( ∧ r j )( u K )) ⊗ w ∗ L = j L/K ( ǫ K ) . Since s = 0, the map (4.4) is just the map ∧ r, U K → ( ∧ r, U K ) ⊗ Z Z [ H ] / A H thatsends u to u ⊗
1, so R Art
L/K is the map( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ −→ ( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ ⊗ Z [ H ] / A H that sends u ⊗ w to u ⊗ w ⊗
1. Hence by (5.8) and (5.9) we haveTw
L/K ( ǫ L ) ≡ j L/K ( ǫ K ) ⊗ j L/K ⊗ R Art
L/K ( ǫ K ))in ( ∧ r, U K ) ⊗ Γ ∧ r W ∗ K,S ′ ⊗ Z Z [ H ] / A H , which is Conjecture 5.2(ii). (cid:3) Proposition 5.10. If L = K , then Conjecture 5.2 is true.Proof. If S ′′ = S ′ , then this follows from Proposition 5.7. If S ′′ = S ′ then for everycharacter χ of Γ, we have r ( χ, S ) ≥ | S ′′ | > | S ′ | = r , so ρ L,r = 1 in Definition 3.4and by definition ǫ L = ǫ L,S,T,S ′ = 0. Further, we have A H = 0 in this case, so R Art
K/K = 0 and Conjecture 5.2 holds. (cid:3) Order of vanishing
Fix a number field k , and a set S ′ of archimedean places of k . Let r := | S ′ | .Let T be a finite set of primes of k , containing at least one prime not dividing 2,and containing primes of at least two different residue characteristics if S ′ containsno real places. (This ensures that an extension of k in which all places in S ′ splitcompletely has no roots of unity congruent to one modulo all primes in T .)For example (perhaps the most interesting example), k could be a totally realfield and S ′ the set of all archimedean places, in which case r = [ k : Q ].Fix a finite abelian extension K of k such that all places in S ′ split completelyin K/k , and all places in T are unramified in K/k . Fix a finite set S of places of K disjoint from T , containing all archimedean places, all primes ramifying in K/k ,and at least one place not in S ′ . Let P be the set of all primes of k not in S ∪ T that split completely in K/k , and let N be the set of all squarefree products ofprimes in P .For every q ∈ P suppose that K ( q ) is a finite abelian extension of k containing K ,such that K ( q ) /K is totally tamely ramified above q and unramified everywhereelse, and all places above S ′ split completely in K ( q ) /K . (For example, if K contains the Hilbert class field of k then we could take K ( q ) to be the compositumof K with the ray class field of k modulo q .) If n ∈ N define K ( n ) to be thecompositum of the fields K ( q ) for q dividing n . Ramification considerations showthat all the K ( q ) are linearly disjoint over K , so if we define H ( n ) := Gal( K ( n ) /K )then H ( n ) = Y q | n H ( q )and if m | n we can view H ( m ) both as a quotient and a subgroup of H ( n ). Let π m : H ( n ) ։ H ( m ) ֒ → H ( n ) , denote the projection map. Let S ( n ) := S ∪ { q : q | n } and S ′ ( n ) := S ′ ∪ { q : q | n } . Assume for the rest ofthis section that the generalized Stark conjecture St( K ( n ) /k, S ( n ) , T, S ′ ) holds forevery n ∈ N , with an element ǫ n := ǫ K ( n ) ,S ( n ) ,T,S ′ ∈ ∧ r, U K ( n ) ,S ( n ) ⊗ ∧ r W ∗ K ( n ) ,S ′ . Lemma 6.1. If d | n then X γ ∈ H ( n / d ) γǫ n = (cid:16)Q q | ( n / d ) (1 − Fr − q ) (cid:17) j K ( n ) /K ( d ) ( ǫ d ) . Proof.
This follows from [R1, Proposition 6.1] and the definition (Lemma 4.9) of j K ( n ) /K ( d ) . (cid:3) Let ν ( n ) denote the number of prime factors of n . Lemma 6.2.
We have X γ ∈ H ( n ) γǫ n ⊗ Q q | n ( π q ( γ ) − X d | n X γ ∈ H ( d ) γ j K ( n ) /K ( d ) ( ǫ d ) ⊗ γ Q q | ( n / d ) ( π d (Fr q ) − in ∧ r, U K ( n ) ,S ( n ) ⊗ ∧ r W ∗ K ( n ) ,S ′ ⊗ Z [ H ( n )] .Proof. Expanding gives X γ ∈ H ( n ) γǫ n ⊗ Q q | n ( π q ( γ ) −
1) = X γ ∈ H ( n ) X d | n ( − ν ( n / d ) γǫ n ⊗ π d ( γ ) . For every d dividing n , using Lemma 6.1 we have X γ ∈ H ( n ) γǫ n ⊗ π d ( γ ) = X γ ∈ H ( d ) (cid:0) γ P h ∈ H ( n / d ) hǫ n (cid:1) ⊗ γ = X γ ∈ H ( d ) γ Q q | ( n / d ) (1 − Fr − q ) j K ( n ) /K ( d ) ( ǫ d ) ⊗ γ = X γ ∈ H ( d ) γ j K ( n ) /K ( d ) ( ǫ d ) ⊗ γ Q q | ( n / d ) (1 − π d (Fr q )) . Combining these identities proves the lemma. (cid:3)
Theorem 6.3.
Suppose that the Stark conjecture
St( K ( n ) /k, S ( n ) , T, S ′ ) holds forevery n ∈ N . Then for every n ∈ N , we have Tw K ( n ) /K ( ǫ n ) ∈ ∧ r, U K ( n ) ,S ( n ) ⊗ ∧ r W ∗ K ( n ) ,S ′ ⊗ A ν ( n ) H ( n ) . In other words, Conjecture 5.2(i) holds for ( K ( n ) /K/k, S ( n ) , T, S ′ , S ′ ( n )) .Proof. The proof, by induction on ν ( n ), is essentially the same as that of [D, Lemma8.1]. In the equality of Lemma 6.2, every term except possibly Tw K ( n ) /K ( ǫ n ) (thesummand on the right with d = n ) lies in ∧ r, U K ( n ) ,S ( n ) ⊗ ∧ r W ∗ K ( n ) ,S ′ ⊗ A ν ( n ) H ( n ) byour induction hypothesis. Therefore Tw K ( n ) /K ( ǫ n ) does as well. (cid:3) EFINED CLASS NUMBER FORMULAS FOR G m The case K = k In this section we consider the case K = k . Let S ′ , S , T , N , k ( q ), k ( n ), H ( n ), S ( n ), S ′ ( n ) be as in §
6, and recall that r := | S ′ | . We will show under mildhypotheses that Conjecture 5.2 holds in this case (with both sides of Conjecture5.2(ii) equal to zero). This is needed for the proof of Theorem 10.7 below, becauseour general techniques only work for nontrivial characters of K/k . Lemma 7.1.
Suppose that S ′ does not contain all archimedean places of k . Then ǫ k ( n ) ,S ( n ) ,T,S ′ = 0 for every n = 1 .Proof. Let w be an archimedean place not in S ′ . By definition k ( n ) /k is unramifiedoutside of n , so w splits completely in k ( n ) /k . Hence if n = 1 then ǫ k ( n ) ,S ( n ) ,T,S ′ = 0by Lemma 3.6. (cid:3) Theorem 7.2.
Suppose n ∈ N and Conjecture St( k ( n ) /k ) holds. If | S − S ′ | ≥ ,or if S ′ does not contain all archimedean places of k , then Conjecture 5.2 holds for ( k ( n ) /k/k, S ( n ) , T, S ′ , S ′ ( n )) .Proof. Conjecture 5.2(i) holds by Theorem 6.3, and Conjecture 5.2(ii) holds when n = 1 by Proposition 5.10. To prove the theorem we will show that for every n = 1,Tw k ( n ) /k ( ǫ k ( n ) ,S ( n ) ,T,S ′ ) ∈ ∧ r, U k ( n ) ,S ( n ) ⊗ ∧ r W ∗ k ( n ) ,S ′ ⊗ A ν ( n )+1 H ( n ) , (7.3) R Art k ( n ) /k ( ǫ k,S ( n ) ,T,S ′ ( n ) ) ∈ ∧ r, U k ( n ) ,S ( n ) ⊗ ∧ r W ∗ k ( n ) ,S ′ ⊗ A ν ( n )+1 H ( n ) . (7.4)Suppose first that | S − S ′ | ≥
2. Then ǫ k,S ( n ) ,T,S ′ ( n ) = 0 by Lemma 3.6, so (7.4)holds. If k has an archimedean place not in S ′ , then ǫ k ( n ) ,S ( n ) ,T,S ′ ( n ) = 0 for n = 1by Lemma 7.1, so (7.3) holds. If not, then S contains two nonarchimedean primes;call one of them v and let S := S − { v } . Since v does not divide n and S is stillstrictly larger than S ′ , all the hypotheses of Conjecture St( k ( n ) /k, S ( n ) , T, S ′ ) aresatisfied, so by Theorem 6.3 we have(7.5) Tw k ( n ) /k ( ǫ k ( n ) ,S ( n ) ,T,S ′ ) ∈ ∧ r, U k ( n ) ,S ( n ) ⊗ ∧ r W ∗ k ( n ) ,S ′ ⊗ A ν ( n ) H ( n ) . It follows directly from the defining properties (see for example [R1, Proposition3.6]) that ǫ k ( n ) ,S ( n ) ,T,S ′ = (1 − Fr − v ) ǫ k ( n ) ,S ( n ) ,T,S ′ , so using (7.5)Tw k ( n ) /k ( ǫ k ( n ) ,S ( n ) ,T,S ′ ) = Tw k ( n ) /k ( ǫ k ( n ) ,S ( n ) ,T,S ′ )(1 − Fr − v ) ∈ ∧ r, U k ( n ) ,S ( n ) ⊗ ∧ r W ∗ k ( n ) ,S ′ ⊗ A ν ( n )+1 H ( n ) . This is (7.3).Now suppose that S ′ does not contain all archimedean places of k . By Lemma7.1 we have ǫ k ( n ) ,S ( n ) ,T,S ′ = 0 for every n = 1, so (7.3) holds. If S contains anonarchimedean place then | S − S ′ | ≥
2, and we are in the case treated above. Sowe may assume that S is the set of all archimedean places. Let S ′ = { v , . . . , v r } and n = Q si =1 q i . For 1 ≤ i ≤ s define η i : U k,S ( n ) → A H ( n ) / A H ( n ) to be the mapgiven by the local Artin symbol η i ( u ) := [ u, k ( n ) q i /k q i ] − k ( n ) q i is the completion of k ( n ) at a prime above q i . Fix an expression ǫ k,S ( n ) ,T,S ′ ( n ) = ( u ∧ · · · ∧ u r + s ) ⊗ ( v ∗ ∧ · · · ∧ v ∗ r ∧ q ∗ ∧ · · · ∧ q ∗ s ) with u i ∈ U k,S ( n ) (we have ∧ r + s, U k,S ( n ) = ∧ r + s U k,S ( n ) since Z [Γ] = Z ). Thenconcretely (ignoring the sign, which will not be important)(7.6) R Art k ( n ) /k ( ǫ k,S ( n ) ,T,S ′ ( n ) ) = ± ( η ∧ · · · ∧ η s )( u ∧ · · · ∧ u r + s ) ⊗ ( v ∗ ∧ · · · ∧ v ∗ r ) . In A H ( n ) / A H ( n ) , using the reciprocity law of global class field theory, we have forevery u ∈ U k,S ( n ) s X i =1 η i ( u ) = (cid:18)Y q | n [ u, k ( n ) q /k q ] (cid:19) − Y w ∤ n [ u, k ( n ) w /k w ] − − . If w is nonarchimedean and does not divide n , then u is a unit at w and w is unram-ified in k ( n ) /k , so [ u, k ( n ) w /k w ] = 1. If w is archimedean, then w splits completelyin k ( n ) /k , so again [ u, k ( n ) w /k w ] = 1. Thus P si =1 η i : U k,S ( n ) → A H ( n ) / A H ( n ) isthe zero map, and we conclude using (7.6) that R Art k ( n ) /k ( ǫ k,S ( n ) ,T,S ′ ( n ) ) = ± ( η ∧ · · · ∧ η s )( u ∧ · · · ∧ u r + s ) ⊗ ( v ∗ ∧ · · · ∧ v ∗ r )= ± ( η ∧ · · · ∧ η s − ∧ ( P i η i ))( u ∧ · · · ∧ u r + s ) ⊗ ( v ∗ ∧ · · · ∧ v ∗ r ) = 0 . Thus (7.4) holds in this case as well, and the theorem follows. (cid:3) Connection with Euler systems
Let
K/k , S ′ , S , T , P , N , K ( q ), K ( n ), S ( n ), S ′ ( n ) be as in §
6, and let Γ =Gal(
K/k ). Recall that r := | S ′ | .We assume further (by shrinking K ( q ) if necessary) that [ K ( q ) : K ] is prime to[ K : k ] for every q ∈ P . It follows that for every q there is a unique extension k ( q ) /k ,totally ramified at q and unramified elsewhere, such that K ( q ) = Kk ( q ). Then if k ( n ) denotes the compositum of the k ( q ) for q dividing n , we have K ( n ) = Kk ( n )for every n ∈ N , and(8.1) Gal( K ( n ) /k ) ∼ = Γ × H ( n ) . Since all archimedean places split completely in k ( q ) /k for every q , every v ∈ S ′ splits completely in K ( n ) /k for every n . Hence all hypotheses of ConjectureSt( K ( n ) /k, S ( n ) , T, S ′ ) are satisfied.Fix an ordering v , . . . , v r of the places in S ′ , and for each i choose a place w i ofthe algebraic closure ¯ k above v i . Then for every n , the element w ∗ n := ( w | K ( n ) ) ∗ ∧ · · · ∧ ( w r | K ( n ) ) ∗ is a generator of the free, rank-one Z [Gal( K ( n ) /k )]-module ∧ r W ∗ K ( n ) ,S ′ . When n = 1 we will write w ∗ K instead of w ∗ . Definition 8.2.
As in §
6, for every n ∈ N we define ǫ n := ǫ K ( n ) ,S ( n ) ,T,S ′ ∈ ( ∧ r, U K ( n ) ,S ( n ) ) ⊗ ∧ r W ∗ K ( n ) ,S ′ to be the element predicted by Conjecture St( K ( n ) /k, S ( n ) , T, S ′ ), and we define ξ n ∈ ∧ r, U K ( n ) ,S ( n ) ⊂ ∧ r U K ( n ) ,S ( n ) ⊗ Q to be the unique element satisfying ξ n ⊗ w ∗ n = ǫ n . EFINED CLASS NUMBER FORMULAS FOR G m Proposition 8.3. If m , n ∈ N , and m | n , then N rK ( n ) /K ( m ) ξ n = Y q | ( n / m ) (1 − Fr − q ) ξ m . Proof.
This is [R1, Proposition 6.1]. (cid:3)
By (8.1), for every n ∈ N we can view any Gal( K ( n ) /k )-module as a Γ-module.Fix a rational prime p , not lying below any prime in T , and not dividing [ K : k ].Fix also a character χ : Γ → ¯ Q × p . Let O := Z p [ χ ], the extension of Z p generated bythe values of χ . Since p ∤ [ K : k ], the order of χ is prime to p so O is unramifiedover Z p . If M is a Z [Γ]-module, we let M χ be the submodule of M ⊗ Z O on whichΓ acts via χ . If m ∈ M , then(8.4) m χ := 1[ K : k ] X γ ∈ Γ m γ ⊗ χ − ( γ ) ∈ M χ is the projection of m into M χ .Let M χ := Z p (1) ⊗ χ − denote a free O -module of rank one on which G k actsvia χ − times the cyclotomic character. Proposition 8.5.
For every n ∈ N , Kummer theory gives Galois-equivariant iso-morphisms ( K ( n ) × ) χ ∼ = H ( k ( n ) , M χ ) , and if q is a prime of k (( K ⊗ k k q ) × ) χ ∼ = H ( k q , M χ ) . Proof.
This is a standard calculation; see for example [MR1, § § (cid:3) Theorem 8.6.
Suppose that r = 1 , and Conjecture St( K ( n ) /k, S ( n ) , T, S ′ ) holdsfor every n ∈ N . Let c n ∈ H ( k ( n ) , M χ ) denote the image of ξ χ n under the Kummermap of Proposition 8.5. Then the collection { c n : n ∈ N } is an Euler system for the G k -representation M χ in the sense of [MR1, Definition3.2.2] or [R2, § .Proof. It follows from Proposition 8.3 and (8.4) that if m , n ∈ N and m | n , then N K ( n ) /K ( m ) ξ χ n = Y q | ( n / m ) (1 − Fr − q ) ξ χ m . Translated to the elements c n and c m , this is the defining property of an Eulersystem for M χ . (Note that by the definition of N in §
6, we have χ ( q ) = 1 if q | n .) (cid:3) Remark 8.7.
For general r ≥
1, the collection { c n : n ∈ N } is not necessarilyan Euler system in the sense of [PR, Definition 1.2.2], because the elements c n liein ∧ r, H ( k ( n ) , M χ ) rather than ∧ r H ( k ( n ) , M χ ). This suggests that one mightwant to relax the definition of Euler system to allow elements to lie in the largerlattice. Connection with Stark systems
Let K ( n ) /K/k , Γ, S ′ , r , S , T , P , N , S ( n ), S ′ ( n ), χ and M χ be as in § §
8. For n ∈ N let ν ( n ) denote the number of primes dividing n . We continue tosuppose that [ K ( q ) : K ] is prime to [ K : k ] for every q ∈ P , and we now supposein addition that(9.1) p ∤ [ K : k ] Y λ ∈ T K ( N λ − A denote the ring of integers of K , and for every n ∈ N let A S ( n ) denote the S ( n )-integers of KA S ( n ) := { x ∈ K : ord λ ( x ) ≥ λ / ∈ S ( n ) K } . Then U K,S ( n ) = { u ∈ A × S ( n ) : u ≡ λ ) for every λ ∈ T K } . Lemma 9.2.
For every n ∈ N we have p ∤ [ A × S ( n ) : U K,S ( n ) ] Proof.
Reduction gives an injection A × S ( n ) /U K,S ( n ) ֒ → ⊕ λ ∈ T K ( A/λ ) × , so the lemmafollows from our assumption (9.1). (cid:3) Lemma 9.3.
For every n ∈ N we have ( ∧ r + ν ( n ) , U K,S ( n ) ) χ = ∧ r + ν ( n ) U χK,S ( n ) .Proof. By our choice of T , the group U K,S ( n ) is torsion-free. Since p ∤ [ K : k ], wehave [ K : k ] ∈ O × , so U K,S ( n ) ⊗ O is a projective O [Γ]-module. It now follows fromLemma A.4 that ∧ r + ν ( n ) , U K,S ( n ) ⊗ O = ∧ r + ν ( n ) U K,S ( n ) ⊗ O . Taking χ -components proves the lemma. (cid:3) Define N p = { n ∈ N : n is prime to p } . For n ∈ N p recall that H ( n ) := Gal( K ( n ) /K ), and A H ( n ) ⊂ O [ H ( n )] is the aug-mentation ideal. Define an ideal I n ⊂ O by I n := X q | n ([ k ( q ) : k ] O )(with the convention I = 0). Let W K, n denote the free abelian group on the set ofprimes of K dividing n , so W K,S ′ ( n ) = W K,S ′ ⊕ W K, n and(9.4) ∧ r + ν ( n ) W ∗ K,S ′ ( n ) = ∧ ν ( n ) W ∗ K, n ⊗ ∧ r W ∗ K,S ′ . Definition 9.5.
For every n ∈ N p , define Y n := ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( n ) ( W ∗ K,S ′ ( n ) ) χ ⊗ ( O /I n ) . If m | n , we define a map Ψ n , m : Y n −→ Y m ⊗ ( O /I n )as follows. Fix a prime factorization n / m = q · · · q t and for each i fix a prime Q i of K above q i . Define ψ i ∈ U ∗ K,S ( n ) by ψ i ( u ) = P γ ∈ Γ ord Q i ( u γ ) γ − . By DefinitionA.1 we get a map ψ ∧ · · · ∧ ψ t : ∧ r + ν ( n ) U χK,S ( n ) −→ ∧ r + ν ( m ) U χK,S ( n )EFINED CLASS NUMBER FORMULAS FOR G m and by [R1, Lemma 5.1] or [MR4, Proposition A.1] the image of this map is con-tained in ∧ r + ν ( m ) U χK,S ( m ) . Further, viewing Q ∧ · · · ∧ Q t as a generator of ∧ t W n / m the map(9.6) ( ψ ∧ · · · ∧ ψ t ) ⊗ ( Q ∧ · · · ∧ Q t ): ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ t ( W ∗ n / m ) χ −→ ∧ r + ν ( m ) U χK,S ( m ) is independent of the choice of the Q i and the order of the q i . Now we define Ψ n , m to be the composition Y n = ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( n ) ( W ∗ K,S ′ ( n ) ) χ ⊗ ( O /I n ) ∼ −−→ ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( m ) ( W ∗ K,S ′ ( m ) ) χ ⊗ ∧ ν ( n / m ) ( W ∗ n / m ) χ ⊗ ( O /I n ) −→ ∧ r + ν ( m ) U χK,S ( m ) ⊗ ∧ ν ( m ) ( W ∗ K,S ′ ( m ) ) χ ⊗ ( O /I n ) = Y m ⊗ ( O /I n ) , where the last map is induced by (9.6). Note that Ψ n , m is the map Φ of [R1, § ǫ χ n ∈ Y n , where ǫ n is the element of Definition 8.2predicted by Conjecture St( K ( n ) /k, S ( n ) , T, S ′ ). The following lemma allows us toapply the results of [MR4] to the family of Y n . Lemma 9.7.
The modules Y n and the maps Ψ n , m defined above are the same asthe Y n and Ψ n , m of [MR4, Definition 7.1] for the Galois representation M χ .Proof. The proof is an exercise, using the natural Kummer theory isomorphisms( K × ) χ ∼ = H ( k, M χ ) and (( K ⊗ k v ) × ) χ ∼ = H ( k v , M χ ) for places v of k (Proposition8.5), along with Lemma 9.2. (cid:3) Definition 9.8.
As in [MR4, Definition 7.1] we say that a collection { σ n ∈ Y n : n ∈ N p } is a Stark system of rank r ifΨ n , m ( σ n ) = σ m ⊗ ∈ Y m ⊗ ( O /I n ) whenever m | n ∈ N p .Let SS r ( M χ ) denote the O -module of Stark systems of rank r .Suppose for the rest of this section that Conjecture St( K/k, S ( n ) , T, S ′ ( n )) holdsfor every n ∈ N . Recall that w ∗ K is the generator of ∧ r W ∗ K,S ′ fixed at the beginningof §
8, and denotes the trivial character of Γ. Definition 9.9.
For n ∈ N let δ n ∈ ( ∧ r + ν ( n ) U K,S ( n ) ) ⊗ ∧ ν ( n ) ( W ∗ K, n ) be the uniqueelement such that δ n ⊗ w ∗ K := ǫ K,S ( n ) ,S ′ ( n ) ∈ ( ∧ r + ν ( n ) U K,S ( n ) ) ⊗ ∧ r + ν ( n ) W ∗ K,S ′ ( n ) is the element predicted by Conjecture St( K/k, S ( n ) , T, S ′ ( n )), using the identifi-cations of Lemma 9.3 and (9.4). Then δ χ n ⊗ ∈ ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( n ) ( W ∗ K,S ′ ( n ) ) χ ⊗ ( O /I n ) = Y n , and we denote by δ χ the collection { δ χ n ⊗ ∈ Y n : n ∈ N p } . Proposition 9.10.
We have δ χ ∈ SS r ( M χ ) , i.e., δ χ is a Stark system of rank r .Proof. If n ∈ N and m | n , then Ψ n , m ( δ χ n ⊗
1) = δ χ m ⊗ (cid:3) Let r ( χ, S ) be as in Definition 3.4. Lemma 9.11. (i) If r ( χ, S ) > r , then δ χ n = 0 for every n ∈ N . (ii) If r ( χ, S ) = r , then δ χ n is a nonzero element of the free, rank-one O -module ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( n ) ( W ∗ K, n ) χ .Proof. The Z [Γ]-module W ∗ K, n is free of rank ν ( n ). By the basic properties ofConjecture St( K/k, S ( n ) , T, S ′ ( n )) we have δ χ n = 0 ⇐⇒ r ( χ, S ( n )) = r + ν ( n ) ⇐⇒ r ( χ, S ) = r, and if these equivalent conditions hold then U χK,S ( n ) is free of rank r + ν ( n ) over O .The lemma follows. (cid:3) The case r = 1Keep the setting and notation of the previous two sections. In this section wewill prove (Theorem 10.7) a part of Conjecture 5.2(ii) when r = 1. The idea of theproof is as follows.The Stark system δ χ of § M χ . When r = 1, the Euler system of Stark elements of Theorem8.6 also gives rise (via an explicit construction) to a Kolyvagin system for M χ .The O -module of Kolyvagin systems for M χ is free of rank one, and the twoKolyvagin systems agree when n = 1 by construction. Hence the two Kolyvaginsystems agree for every n , and unwinding the two explicit constructions showsthat the agreement for n is equivalent to the “( p, χ )-part” of Conjecture 5.2(ii) for( K ( n ) /K/k, S ( n ) , T, S ′ ).As in §
6, if m | n we can view H ( m ) as both a subgroup and a quotient of H ( n ),and π m : H ( n ) ։ H ( m ) ֒ → H ( n ) is the projection map. Definition 10.1. If n ∈ N and d = Q ti =1 q i divides n , let M n , d = ( m ij ) be the t × t matrix with entries in A H ( n ) / A H ( n ) m ij = ( π n / d (Fr q i −
1) if i = j,π q j (Fr q i −
1) if i = j, and define D n , d := det( M n , d ) ∈ A tH ( n ) / A t +1 H ( n ) (this is independent of the ordering of the prime factors of d ). By convention welet D n , = 1. For n ∈ N , let B n denote the cyclic group B n := { Q q | n ( γ q −
1) : γ q ∈ H ( q ) } ⊂ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . By [MR3, Proposition 4.2], B n is a direct summand of A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) .Let KS r ( M χ ) denote the O -module of Kolyvagin systems of rank r for M χ (with the natural Selmer structure of [MR4, § §
10] (seealso [MR4, § § § r for M χ is a collection { κ n ∈ ∧ r U χK,S ( n ) ⊗ B n : n ∈ N p } satisfying properties that we do not need to review here. We are identifying ⊗ q | n H ( q ) with B n via ⊗ q γ q Q q ( γ q − EFINED CLASS NUMBER FORMULAS FOR G m Definition 10.2.
For n ∈ N let δ χ n ∈ ∧ r + ν ( n ) U χK,S ( n ) ⊗ ∧ ν ( n ) ( W ∗ K, n ) χ be as inDefinition 9.9, and define β St n := X d | n R Art K ( d ) /K ( δ χ d ) · D n , n / d ∈ ∧ r U χK,S ( n ) ⊗ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . Proposition 10.3.
For n ∈ N we have β St n ∈ ∧ r U χK ( n ) ,S ( n ) ⊗ B n , and the collection β St := { β St n : n ∈ N p } is a Kolyvagin system of rank r for M χ .Proof. In the special case where k = Q , S ′ = {∞} , and χ is an even quadraticcharacter, this is [MR3, Theorem 8.7 and Proposition 6.5]. The proof in general issimilar. The general case is also proved by Sano in [S2, §
4] (what we call a Starksystem is called a unit system in [S2]). (cid:3)
For the rest of this section we assume that r = 1, i.e., S ′ consists of a singlearchimedean place. Since r = 1, the Stark unit Euler system of Theorem 8.6 givesrise, via the map of [MR1, Theorem 3.2.4], to a Kolyvagin system of rank one κ St = { κ St n : n ∈ N p } ∈ KS ( M χ ) . (The results of [MR1] are stated only for k = Q , but the proofs in the general caseare the same; see [MR4].) Proposition 10.4.
Suppose n ∈ N p . Under the restriction map K × → K ( n ) × andthe inclusion B n ⊂ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) , with ξ n as in Definition 8.2 we have κ St n X d | n Tw K ( d ) /K ( ξ χ d ) · D n , n / d ∈ U χK ( n ) ,S ( n ) ⊗ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . Proof.
Note that Tw K ( d ) /K ( ξ χ d ) lies in U χK ( n ) ,S ( n ) ⊗ A ν ( d ) H ( n ) / A ν ( d )+1 H ( n ) by Theorem 6.3and Lemma 9.3, and D n , n / d lies in A ν ( n / d ) H ( n ) / A ν ( n / d )+1 H ( n ) by definition.In the special case where k = Q and χ is a real quadratic character, this is [MR3,Theorem 7.2 and Proposition 6.5]. The proof in general is the same. The generalcase also follows from calculations of Sano [S2, § (cid:3) Theorem 10.5. If χ = then for every n ∈ N we have κ St n = β St n .Proof. Let r ( χ, S ) be as in Definition 3.4, and suppose first that r ( χ, S ) = 1. Wehave κ St , β St ∈ KS ( M χ ). Since χ = and K contains no nontrivial p -th roots ofunity by Lemma 9.2, all the hypotheses of [MR1, § KS ( M χ ) is a free O -module of rank one by [MR1, Theorem 5.2.10]. We have β St1 = δ χ = ξ χ = κ St1 by definition, and by Lemma 9.11(ii) this is a nonzero element of the free, rank-one O -module U χK,S . Hence β St = κ St , i.e., κ St n = β St n for every n ∈ N p .Now suppose r ( χ, S ) >
1. By Lemma 9.11(i), we have δ χ n = 0 for every n , so β St n = 0 for every n . Since κ St1 = 0, the finiteness of the ideal class group togetherwith [MR4, Theorem 13.4(iv) and Proposition 5.7] (see also [MR1, Theorem 5.2.12])shows that κ St = 0, i.e., κ St n = 0 for every n ∈ N p .It remains to show that κ St n = β St n ∈ U χK,S ( n ) ⊗ B n when n ∈ N − N p . But theexponent of the cyclic group B n is the greatest common divisor of the | H ( q ) | for q dividing n . If q | p then (since K ( q ) is tamely ramified by definition) H ( q ) has order prime to p . Hence B n has order prime to p if n ∈ N − N p , so B n ⊗ O = 0 and κ St n = β St n = 0. This completes the proof. (cid:3) Theorem 10.6.
Suppose that | S ′ | = 1 , that Conjectures St(
K/k ) and St( K ( n ) /k ) hold for every n , and that at least one of the following holds: (a) χ = , (b) χ = and | S − S ′ | ≥ , (c) χ = and k has more than one archimedean place,Then for every n ∈ N , Tw K ( n ) /K ( ǫ χK ( n ) ,S ( n ) ,T,S ′ ) = R Art K ( n ) /K ( ǫ χK,S ( n ) ,T,S ′ ( n ) ) in U K ( n ) ,S ( n ) ⊗ Gal( K ( n ) /k ) W ∗ K ( n ) ,S ′ ⊗ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . In other words, the ( p, χ ) partof Conjecture 5.2(ii) holds for ( K ( n ) /K/k, S ( n ) , T, S ′ ) .Proof. If χ = , then this follows directly from Theorem 10.5 by induction on n ,using Proposition 10.4 and Definition 10.2 for the induction. If χ = , then this isTheorem 7.2. (cid:3) Let Σ = Σ(
K/k, T ) be the set of primes dividing [ K : k ] Q λ ∈ T K ( N λ − Theorem 10.7.
Suppose that | S ′ | = 1 , that Conjectures St(
K/k ) and St( K ( n ) /k ) hold for every n , and that either k has more than one archimedean place or | S | ≥ .Then Conjecture 5.2(ii) holds for ( K ( n ) /K/k, S ( n ) , T, S ′ ) away from Σ , i.e., forevery p / ∈ Σ the leading term formula holds if we tensor with Z p .Proof. We can apply Theorem 10.6 for every prime p / ∈ Σ, and every character χ of Γ. Summing the conclusion of Theorem 10.6 over all χ gives the equality ofConjecture 5.2(ii) tensored with O . (cid:3) Evidence in the case of general r Keep the notation of the previous sections. When r >
1, the proof of §
10 breaksdown. Namely, the elements ξ n of Definition 8.2 naturally form an Euler system ofrank r , but when r > r . However, using ideas of [R1, §
6] and [B¨uy] we definea family of “projectors” Φ, each of which maps the collection { ξ χ n } to an Eulersystem ξ StΦ of rank one, and maps the rank- r Kolyvagin system β St to a rank-oneKolyvagin system β StΦ . We can associate to ξ StΦ a Kolyvagin system κ StΦ of rank one,and the arguments of §
10 will show that β StΦ = κ StΦ . Unwinding the definitions, thisshows that the Φ-projection of the leading term formula of Conjecture 5.2 holds.For this section we make the extra assumptions that • S contains no primes above p , • k is totally real of degree r and S ′ is the set of its archimedean places, • Leopoldt’s conjecture holds for K .In particular K is totally real and K/k is unramified above p . Definition 11.1.
For every n ∈ N p let V K ( n ) denote the p -adic completion of thelocal units of K ( n ) ⊗ Q p , and V ∗ K ( n ) := Hom Gal( K ( n ) /k ) ( V K ( n ) , Z p [Gal( K ( n ) /k )]). If φ ∈ V ∗ K ( n ) , then ˜ φ will denote the composition˜ φ : U K ( n ) ,S ( n ) −→ V K ( n ) −→ Z p [Gal( K ( n ) /k )] . EFINED CLASS NUMBER FORMULAS FOR G m Define V ∗∞ := lim ←− V ∗ K ( n ) , where the inverse limit is taken with respect to the maps V ∗ K ( nq ) → V ∗ K ( n ) induced by V K ( n ) ⊂ V K ( nq ) , Z p [Gal( K ( nq ) /k )] Gal( K ( nq ) /K ( n )) = Z p [Gal( K ( n ) /k )] . If Φ := φ ∧ . . . ∧ φ r − ∈ ∧ r − V ∗∞ , with φ i ∈ V ∗∞ , let φ i,K ( n ) : V K ( n ) −→ Z p [Gal( K ( n ) /k )]denote the projection of φ i to V ∗ K ( n ) , let˜Φ K ( n ) := ˜ φ ,K ( n ) ∧ · · · ∧ ˜ φ r − ,K ( n ) : ∧ r, U K ( n ) ,S ( n ) −→ U K ( n ) ,S ( n ) be the map of Definition A.1 (combined with Lemmas A.4 and A.5), and let L Φ := ∩ i ker( φ i,K ) ⊂ V K . Using the identification V χK ⊂ ⊕ p | p H ( k p , M χ ) of Proposition 8.5, we define aSelmer structure (see [MR4, Definition 2.1] or [MR1, Definition 2.1.1]) F Φ on M χ by modifying the natural Selmer structure F ur of [MR4, § p ,namely we set ⊕ p | p H F Φ ( k p , M χ ) := L χ Φ ⊂ V χK ⊂ ⊕ p | p H ( k p , M χ ) . Let ξ n ∈ ∧ r, U K ( n ) ,S ( n ) be as in Definition 8.2, and recall the Kolyvagin system β St = { β St n : n ∈ N p } ∈ KS r ( M χ ) of Definition 10.2 and Proposition 10.3. Proposition 11.2.
Suppose
Φ := φ ∧ . . . ∧ φ r − ∈ ∧ r − V ∗∞ . (i) The collection { ˜Φ K ( n ) ( ξ χ n ) ∈ U χK ( n ) ,S ( n ) : n ∈ N p } is an Euler system ofrank one for the representation M χ . (ii) Let κ StΦ = { κ StΦ , n : n ∈ N p } ∈ KS ( M χ ) be the Kolyvagin system of rankone attached to the Euler system of (i) by [MR1, Theorem 3.2.4] . Then κ StΦ ∈ KS ( M χ , F Φ ) , where F Φ is the Selmer structure of Definition 11.1. (iii) The collection β StΦ := { ˜Φ K ( β St n ) : n ∈ N p } is a Kolyvagin system of rankone for ( M χ , F Φ ) .Proof. The first assertion is proved in [R1, Proposition 6.6]. Both (i) and (ii) areproved in [B¨uy, Proposition 2.2 and Theorem 2.19]. Assertion (iii) follows fromProposition 10.3 by direct calculation. (cid:3)
Proposition 11.3.
Suppose
Φ := φ ∧ . . . ∧ φ r − ∈ ∧ r − V ∗∞ and n ∈ N p . Underthe restriction map K × → K ( n ) × and the inclusion B n ⊂ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) , we have κ StΦ , n X d | n Tw K ( d ) /K ( ˜Φ K ( d ) ( ξ χ d )) · D n , n / d ∈ U χK ( n ) ,S ( n ) ⊗ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . Proof.
The proof is similar to Proposition 10.4, or see [S2, § (cid:3) Theorem 11.4.
Suppose
Φ := φ ∧ . . . ∧ φ r − ∈ ∧ r − V ∗∞ . If χ = then for every n ∈ N we have κ StΦ , n = ˜Φ K ( β St n ) .Proof. The proof is similar to that of Theorem 10.5.Let r ( χ, S ) be as in Definition 3.4, and suppose first that r ( χ, S ) = r and φ ,K , . . . , φ r − ,K are Z p -linearly independent. We have κ StΦ , β StΦ ∈ KS ( M χ , F Φ )by Proposition 11.2. The core rank of ( M χ , F Φ ) is one by [B¨uy, Proposition 1.8].All the hypotheses of [MR1, § KS ( M χ , F Φ ) is a free O -module of rank one by [MR1, Theorem 5.2.10]. We have ˜Φ K ( β St1 ) = ˜Φ K ( δ χ ) = ˜Φ K ( ξ χ ) = κ StΦ , bydefinition, and it follows from Lemma 9.11(ii), our assumption on the independenceof the φ i,K , and Leopoldt’s conjecture that this has infinite order in L χ Φ . Hence β StΦ = κ StΦ , i.e., κ StΦ , n = ˜Φ K ( β St n ) for every n ∈ N p .Now suppose that either r ( χ, S ) > r or the φ i,K are linearly dependent. In theformer case Lemma 9.11(i) shows that δ χ n = 0 for every n , and in the latter case˜Φ K = 0, so in either case ˜Φ K ( β St n ) = 0 for every n . Since κ StΦ , = 0, the finiteness ofthe ideal class group together with [MR4, Theorem 13.4(iv) and Proposition 5.7](see also [MR1, Theorem 5.2.12]) and Leopoldt’s conjecture (see [B¨uy, Remark 1.7])shows that κ StΦ = 0, i.e., κ StΦ , n = 0 for every n ∈ N p .It remains to show that κ St n = β St n ∈ U χK,S ( n ) ⊗ B n when n ∈ N − N p . Thisfollows exactly as in the proof of Theorem 10.5, since B n has order prime to p if n ∈ N − N p . This completes the proof. (cid:3) Theorem 11.5.
Suppose that Conjectures
St(
K/k ) and St( K ( n ) /k ) hold for every n , and either χ = or | S − S ′ | ≥ . Then for every Φ := φ ∧ . . . ∧ φ r − ∈ ∧ r − V ∗∞ and every n ∈ N , ˜Φ K ( n ) (Tw K ( n ) /K ( ǫ χK ( n ) ,S ( n ) ,T,S ′ )) = ˜Φ K ( n ) ( R Art K ( n ) /K ( ǫ χK,S ( n ) ,T,S ′ ( n ) )) in U K ( n ) ,S ( n ) ⊗ Gal( K ( n ) /k ) W ∗ K ( n ) ,S ′ ⊗ A ν ( n ) H ( n ) / A ν ( n )+1 H ( n ) . In other words, the ˜Φ K ( n ) -projection of the χ part of Conjecture 5.2(ii) holds for ( K ( n ) /K/k, S ( n ) , T, S ′ ) .Proof. If χ = , then this follows directly from Theorem 11.4 by induction on n ,using Proposition 11.3 and Definition 10.2 for the induction. If χ = , then this isTheorem 7.2. (cid:3) Let Σ = Σ(
K/k, S, T ) be the set of rational primes dividing[ K : k ] Y λ ∈ S − S ′ N λ Y λ ∈ T K ( N λ − . Theorem 11.6.
Suppose that k is totally real, S ′ is the set of all archimedean placesof k , Conjectures St(
K/k ) and St( K ( n ) /k ) hold for every n , Leopoldt’s conjectureholds for K , and | S − S ′ | ≥ . Then for every p / ∈ Σ , every Φ ∈ ∧ r − V ∗∞ , and every n ∈ N , we have ˜Φ K ( n ) (Tw K ( n ) /K ( ǫ K ( n ) ,S ( n ) ,T,S ′ )) = ˜Φ K ( n ) ( R Art K ( n ) /K ( ǫ K,S ( n ) ,T,S ′ ( n ) )) . In other words, for every Φ ∈ ∧ r − V ∗∞ the leading term formula of Conjecture5.2(ii) holds for ( K ( n ) /K/k, S ( n ) , T, S ′ ) after applying ˜Φ K ( n ) .Proof. We can apply Theorem 11.5 for every prime p / ∈ Σ, and every character χ ofΓ. Summing the conclusion of Theorem 11.5 over all χ gives the ˜Φ K ( n ) -projectionof Conjecture 5.2(ii) tensored with O . (cid:3) Appendix A. Exterior algebras and determinants
Let O be an integral domain with field of fractions F , and let R = O [Γ] with afinite abelian group Γ.If M is an R -module, we let M ∗ := Hom R ( M, R ), and M will denote the imageof M in M ⊗ O F . If ρ ∈ R , then M [ ρ ] denotes the kernel of multiplication by ρ in M .Fix for this appendix an R -module M of finite type. EFINED CLASS NUMBER FORMULAS FOR G m Definition A.1. If r ≥
0, then ∧ r M (or ∧ rR M , if we need to emphasize the ring R ) will denote the r -th exterior power of M in the category of R -modules, with theconvention that ∧ M = R . If ψ ∈ M ∗ and r ≥
1, we view ψ ∈ Hom( ∧ r M, ∧ r − M )by ψ ( m ∧ · · · ∧ m r ) := r X i =1 ( − i +1 ψ ( m i )( m ∧ · · · ∧ m i − ∧ m i +1 ∧ · · · ∧ m r ) . If ψ ∈ ∧ s M ∗ with s ≤ r , we view ψ ∈ Hom( ∧ r M, ∧ r − s M ) by( ψ ∧ · · · ∧ ψ s )( m ) := ψ s ◦ ψ s − ◦ · · · ◦ ψ ( m ) . In particular(A.2) ( φ ∧ · · · ∧ φ r )( m ∧ · · · ∧ m r ) = det( φ i ( m j )) . Definition A.3.
For every r ≥
0, define ∧ r, M := { m ∈ ∧ r M : ψ ( m ) ∈ R for every ψ ∈ ∧ r M ∗ } . In other words, ∧ r, M is the dual lattice to ∧ r M ∗ in ∧ r M ⊗ F . Lemma A.4.
We have ∧ r M ⊂ ∧ r, M , with equality if | Γ | ∈ O × or if r = 1 .Proof. The inclusion follows directly from the definition, and for the rest see [R1,Proposition 1.2]. (If | Γ | ∈ O × the equality holds because M is a projective R -module.) (cid:3) Lemma A.5. If m ∈ ∧ r, M and ψ ∈ ∧ s M ∗ with s ≤ r , then ψ ( m ) ∈ ∧ r − s, M .Proof. If ψ ′ ∈ ∧ r − s M ∗ then ψ ′ ( ψ ( m )) = ( ψ ∧ ψ ′ )( m ) ∈ R because m ∈ ∧ r, M ,so ψ ∧ m ∈ ∧ r − s, M by definition. (cid:3) Proposition A.6.
Suppose M is an R -module that is projective as an O -module,and B is an O -module. For every s ≤ r and ρ ∈ F [Γ] , the construction of DefinitionA.1 induces a canonical pairing ( ∧ r, M )[ ρ ] × ∧ r − s Hom R ( M, R ⊗ O B ) −→ ( ∧ s, M )[ ρ ] ⊗ O B ⊗ ( r − s ) . In particular, when s = 0 this pairing takes values in R [ ρ ] ⊗ Z B ⊗ r .Proof. There are natural isomorphismsHom R ( M, R ) ∼ = Hom O ( M, O ) , Hom R ( M, R ⊗ O B ) ∼ = Hom O ( M, B ) . Since M is a projective O -module, the natural map M ∗ ⊗ O B −→ Hom R ( M, R ⊗ O B )is an isomorphism. This isomorphism gives the first map of( ∧ r, M )[ ρ ] × ∧ r − s Hom R ( M,R ⊗ O B ) ∼ −→ ( ∧ r, M )[ ρ ] × ∧ r − s ( M ∗ ⊗ O B n ) −→ ( ∧ r, M )[ ρ ] × ( ∧ r − s M ∗ ) ⊗ O B ⊗ ( r − s ) −→ ( ∧ s, M )[ ρ ] ⊗ O B ⊗ ( r − s ) and the last map comes from Definition A.1, using Lemma A.5.If s = 0, then ∧ , M = R by definition. (cid:3) Remark A.7.
If (for example) m , . . . , m r ∈ M , φ i , . . . , φ r ∈ Hom R ( M, R ⊗ O B ),and s = 0, then the pairing of Proposition A.6 is given by( m ∧ · · · ∧ m r , φ ∧ · · · ∧ φ r ) det( φ i ( m j )) . The content of Proposition A.6 is that this pairing is defined on all of ∧ r, M , notjust on ∧ r M . References [Bur] D. Burns, Congruences between derivatives of abelian L -functions at s = 0. Invent. Math. (2007) 451–499.[B¨uy] K. B¨uy¨ukboduk, Kolyvagin systems of Stark units.
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